Let f be a function so that (below). Which must be true?I. f is continuous...



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Let f be a function so that (below). Which must be true?
I. f is continuous at x=2
II. f is differentiable at x=2
III. The derivative of f is continuous at x=2
(A) I (B) II (C) I and II (D) I and III (E) II and III

Answer & Explanation



Beginner2023-03-14Added 3 answers

(C) Explanation: Noting that a function f is differentiable at a point x 0 if
lim h 0 f ( x 0 + h ) - f ( x 0 ) h = L
the given information effectively is that f is differentiable at 2 and that f ( 2 ) = 5 .Now, looking at the statements:
I: True
A function's differentiability at a point implies its continuity at that point.
II: True
The given information matches the definition of differentiability at x = 2 .
III: False
The derivative of a function is not necessarily continuous, a classic example being g ( x ) = { x 2 sin ( 1 x ) if x 0 0 if x = 0 , which is differentiable at 0 , but whose derivative has a discontinuity at 0 .

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